New bounds on the cohesion of complete-link and other linkage methods for agglomeration clustering
Sanjoy Dasgupta, Eduardo Laber
TL;DR
The paper targets theoretical guarantees for hierarchical clustering by improving diameter-based bounds of complete-linkage in general metric spaces and separating it from single-linkage using the average-diametric optimum OPT$_{AV}(k)$. It develops a partition-based analysis that tracks clusters as they are merged, bounding the diameter of resulting clusters via auxiliary structures (families, forests) and, in a refined version, a dynamic graph to exploit pure clusters more effectively. The authors establish a first bound $\max_C\mathrm{diam}(C) \le k^{\log_2 3}\mathrm{OPT}_{AV}(k)$ and then tighten results to $(2k-2)\mathrm{OPT}_{DM}(k)$ for $k\le4$ and $k^{1.30}\mathrm{OPT}_{DM}(k)$ for $k>4$, while also extending the approach to average-linkage and minimax through a general theorem for linkage methods that align a distance function with a cohesion cost. These contributions sharpen our understanding of the quality of hierarchical clustering and offer practical insights for choosing linkage methods, especially in small-$k$ scenarios where provable guarantees are most informative.
Abstract
Linkage methods are among the most popular algorithms for hierarchical clustering. Despite their relevance the current knowledge regarding the quality of the clustering produced by these methods is limited. Here, we improve the currently available bounds on the maximum diameter of the clustering obtained by complete-link for metric spaces. One of our new bounds, in contrast to the existing ones, allows us to separate complete-link from single-link in terms of approximation for the diameter, which corroborates the common perception that the former is more suitable than the latter when the goal is producing compact clusters. We also show that our techniques can be employed to derive upper bounds on the cohesion of a class of linkage methods that includes the quite popular average-link.